Method and system for performing complex sampling of signals by using two or more sampling channels and for calculating time delays between these channels

ABSTRACT

A method and system for performing complex sampling of signals using two or more sampling channels and calculating time delays between these channels. The system and method are operable to enable complex sampling of a signal in a frequency-domain by predefined-order sampling, including utilizing a sampling channel for converting an analog signal to a corresponding substantially non-delayed digital signal; and transforming the digital signal into a plurality of corresponding frequency-domain substantially non-delayed discrete components; providing additional sampling channels enabling to perform a predefined-order sampling, the predefined-order depending on a number of the additional sampling channels, each additional sampling channel configured to perform a number of stage, giving rise to the multiplied frequency-domain delayed discrete components; and combining the multiplied frequency-domain delayed discrete components with the corresponding frequency-domain substantially non-delayed discrete components, giving rise to an output frequency-domain complex signal.

REFERENCE TO CO-PENDING APPLICATIONS

This application is a Divisional of application Ser. No. 13/103,531filed May 9, 2011. The disclosure of the prior application is herebyincorporated by reference herein in its entirety.

FIELD OF THE INVENTION

The present invention relates to digital signal processing. Moreparticularly, the present invention relates to a method and system forperforming complex sampling of signals by using two or more samplingchannels (second-order sampling or higher) and calculating correspondingtime delays between the two or more sampling channels.

DEFINITIONS, ACRONYMS AND ABBREVIATIONS

Throughout this specification, the following definitions are employed:

Signal Sampling: is the process of converting a signal (e.g., thatcontinuously varies in time or space) into a numeric sequence (e.g.,having discrete values in time or space). It should be noted that asampler is, generally, a system/device or operation(s) that enablesextracting (producing) one or more samples from a signal. A theoreticalideal sampler produces samples equivalent to the instantaneous value ofthe continuous signal at one or more desired points in time or space.

Complex Sampling: is a sampling, in which an input signal is sampled,for example, by two samplers (sampling channels) that are shifted byninety degrees, each relative to another. The output signal of the abovesampling is a complex signal.

Complex Signal: is a signal consisting of real and imaginary parts. Forexample, if a complex signal is denoted X(t), thenX(t)=x_(real)(t)+i·x_(imaginary)(t), wherein i=√{square root over (−1)}.It should be noted that in actual physical systems, signals x_(real)(t)and x_(imaginary)(t) are both real, but are called the “real” and“imaginary” parts. The multiplier i is used to help define anoperation(s) between different signals.

FFT: is an acronym for Fast Fourier Transform, which is an efficientalgorithm to compute Discrete Fourier Transform (DFT) and its inverse.There are many distinct FFT algorithms in the art, involving a widerange of mathematic calculations, from simple complex-number arithmeticto group theory and number theory. Generally, the output of the FastFourier Transform is called the FFT spectrum.

FFT Bin: is a single frequency component of the FFT spectrum.

BACKGROUND OF THE INVENTION

The subject matter of signal sampling is widely known in the prior art.Generally, it relates to digital signal processing and has highrelevance in a variety of fields, such as communication, electronics,medicine, electro-optics, and many others. For example, in radiocommunication, sampling a signal and obtaining sufficient signalattenuation, while demodulating the desired signal from radiofrequencies as close as possible to the baseband, is one of the maintasks. According to the commonly known Nyquist-Shannon sampling theorem,which is well known in the field of information theory, and inparticular, in the field of digital signal processing andtelecommunications, an analog signal that has been sampled can be fullyreconstructed from the samples if the sampling frequency F_(S) exceeds2B samples per second (2B is a Nyquist rate that is the minimum samplingrate required to avoid aliasing), where B is the bandwidth of theoriginal signal, i.e. F_(S)>2B or F_(S)/2>B (half of the sampling rateis larger than the signal bandwidth). However, the above theorem isvalid when the signal frequency range does not contain whole multiplesor half-multiples of the sampling rate (sampling frequency).

It should be noted that signals that are used in many applications are,in many cases, band limited to a predefined frequency interval, and thusthese signals are called bandpass signals. A uniform sampling theoremfor bandpass signal is known from the prior art, and its analysis isusually based on the time frequency equivalence. Thus, for example, A.W. Kohlenberg proposed the second order sampling for a bandpass signal(in the article titled “Exact interpolation of band-limited functions”,published in the journal of Applied Physics, in 1953, issue 24(12),pages 1432-1436), which is considered to be the simplest case ofnon-uniform sampling where two uniform sampling sequences areinterleaved. Second order sampling allows the theoretical minimalsampling rate of two-times bandwidth, in the form of an average rate, tobe applied independent of the band position. In second order sampling,when the delay τ between two or more samplers is properly predefined,the signal can be fully reconstructed (e.g., by performing signalinterpolation) even when the signal frequency range contains wholemultiples or half-multiples of the sampling frequency.

FIG. 1A schematically illustrates a conventional interpolation system100 of second order sampling, according to the prior art. In FIG. 1A,the input signal X(t) (t is a time parameter) passes through twoAnalog-to-Digital (A/D) converters 105′ and 105″ with a predefined timedelay τ between them. Then, the converted signals X₁(l) and X₂(l) areinputted into interpolation filters 110′ and 110″, respectively, forperforming signal interpolation, which includes digital to analogconversion. After that, the resulting interpolated signals are summedtogether, giving rise to the output signal Y(l), and in turn Y(t).

It should be noted that second order sampling and its limitations arewell-known in the prior art, and this issue is discussed in theliterature. For example, R. G. Vaughan et al., in the article titled“The Theory of Bandpass Sampling” published in the “IEEE Transactions onSignal Processing” journal (volume 39, number 2, pp. 1973-1984,September 1991), discusses sampling of bandpass signals with respect toband position, noise considerations, and parameter sensitivity,presenting acceptable and unacceptable sample rates with specificdiscussion of the practical rates which are non-minimum. According toVaughan et al., the construction of a bandpass signal from second-ordersamples depends on sampling factors and the relative delay between theuniform sampling streams. For another example, M. Valkama et al., in thearticle titled “A Novel Image Rejection Architecture for QuadratureRadio Receivers” published in the “IEEE Transactions on Circuits andSystems” journal (volume 51, number 2, pp. 61-68, February 2004),presents a novel structure for obtaining an image-free basebandobservation of the received bandpass signal by utilizing I/Q(Inphase/Quadrature) signal processing. The phase difference between Iand Q branches is approximated by a relative time delay of one quarterof the carrier cycle. Also, Valkama et al. presents and analyzes ananalog delay processing based model, and then determines the obtainableimage rejection of the delay processing. In addition, Valkama et al. inanother article titled “Second-Order Sampling of Wideband Signals”,published in the “IEEE International Symposium on Circuits and Systems”journal (volume 2, pp. 801-804, May 2001), discusses and analyzes thesecond-order sampling based digital demodulation technique. According toValkama et al., the modest image rejection of the basic second-ordersampling scheme is improved to provide sufficient demodulationperformance also for wideband receivers. Further, for example, H. Yonget al. in the article titled “Second-Order Based Fast Recovery ofBandpass Signals”, published in the “International Conference on SignalProcessing Proceedings” journal (volume 1, pp. 7-10, 1998), discussesfast recovery and frequency-differencing of real bandpass signals basedon second-order sampling. According to H. Yong et al., by usingsecond-order sampling, the sampling rate can be lowered to thebandwidth. Although the spectrum of the two interleaved sampling streamsare aliasing, it is possible to reconstruct the original orfrequency-differencing bandpass signal.

Further, it should be noted that the conventional complex signalprocessing is also used in processing schemes where an input signal isbandpass in its origin, and is to be processed in a lowpass form. Thisnormally requires two-channel processing in quadrature channels toremove an ambiguity as to whether a signal is higher or lower than thebandpass center frequency. The complex signal processing can be extendedto the digital signal processing field, and the processed signal can befirst mixed to zero-center frequency in two quadrature channels, thenfiltered to remove the high frequency mixing products, and after thatdigitized by a number of A/D (Analog-to-Digital) converters.

According to the prior art, FIG. 1B schematically illustrates aconventional complex sampling system 160, in which an input signal issampled in two sampling channels 150′ and 150″, while shifting the phaseby ninety degrees. At the output of such a system, a complex signal isobtained, said signal having a real part Re{X(l)} and an imaginary partIm{X(l)}, wherein parameter l represents a series of discrete values.Filters 151, 152′ and 152″ are used to filter the undesired frequencyrange (in a time domain) of input signals X(t), X₁′(t) and X₂′(t),respectively.

U.S. Pat. No. 5,099,194 discloses an approach to extending the frequencyrange uses non-uniform sampling to gain the advantages of a highsampling rate with only a modest increase in the number of samples. Twosets of uniform samples with slightly different sampling frequency areused. Each set of samples is Fourier transformed independently and thefrequency of the lowest aliases determined. It is shown that knowledgeof these two alias frequencies permits unambiguous determination of thesignal frequency over a range far exceeding the Nyquist frequency,except at a discrete set of points.

U.S. Pat. No. 5,099,243 presents a technique for extending the frequencyrange which employs in-phase and quadrature components of the signalcoupled with non-uniform sampling to gain the advantages of a highsampling rate with only a small increase in the number of samples. Byshifting the phase of the local oscillator by 90 degrees, a quadratureIF signal can be generated. Both in-phase and quadrature components aresampled and the samples are combined to form a complex signal. When thissignal is transformed, only one alias is obtained per periodicrepetition and the effective Nyquist frequency is doubled. Two sets ofcomplex samples are then used with the slightly different samplingfrequency. Each set is independently Fourier transformed and thefrequency of the lowest aliases permits unambiguous determination of thesignal frequency over a range far exceeding the Nyquist frequency.

U.S. Pat. No. 5,109,188 teaches a technique for extending the frequencyrange which employs a power divider having two outputs, one output beingsupplied to a first Analog-to-Digital (A/D) converter, and the otheroutput being supplied via a delay device to a second A/D converter. Aprocessor receives the outputs of the two A/D converters. In operation,the input signal is subjected to a known delay and both original anddelayed signals are sampled simultaneously. Both sampled signals areFourier transformed and the phase and amplitudes calculated. The phasedifference between the original and delayed signals is also calculated,and an approximation to the true frequency for each peak observed in theamplitude spectrum is estimated.

Based on the above observations, there is a continuous need in the artto provide a method and system configured to perform complex sampling ofsignals by using two or more sampling channels (second-order sampling orhigher) and enabling operating with a signal bandwidth that can be equalto the sampling frequency (or to higher multiples of the samplingfrequency). In addition, there is a need in the art to provide a methodand system for performing signal processing by using second order (orhigher order) sampling, in a frequency domain, without consideringwhether the signal frequency range contains whole multiples orhalf-multiples of the sampling frequency. Further, there is a continuousneed in the prior art to enable calculating corresponding time delaysbetween the two or more sampling channels in a relatively accuratemanner.

SUMMARY OF THE INVENTION

The present invention relates to a method and system for performingcomplex sampling of signals by using two or more sampling channels(second-order sampling or higher) and calculating corresponding timedelays between the two or more sampling channels.

A system is configured to perform a complex sampling of a signal in afrequency-domain by means of a predefined-order sampling, said systemcomprising:

-   -   a) a sampling channel comprising:        -   a.1. at least one analog-to-digital converter configured to            convert an analog signal to a corresponding substantially            non-delayed digital signal; and        -   a.2. at least one frequency-domain discrete transformation            unit for transforming said digital signal to a plurality of            corresponding frequency-domain substantially non-delayed            discrete components;    -   b) one or more additional sampling channels enabling to perform        a predefined-order sampling, the predefined-order depending on a        number of said one or more additional sampling channels, each        additional sampling channel comprising:        -   b.1. at least one delay unit configured to delay an analog            signal by a predefined value, giving rise to a delayed            analog signal;        -   b.2. at least one analog-to-digital converter configured to            convert said delayed analog signal to a corresponding            delayed digital signal;        -   b.3. at least one frequency-domain discrete transformation            unit for transforming said delayed digital signal to a            plurality of frequency-domain delayed discrete components;        -   b.4. at least one data unit configured to provided one or            more corresponding coefficients for each frequency-domain            delayed discrete component; and        -   b.5. at least one multiplication unit configured to multiply            said one or more corresponding coefficients with said each            corresponding frequency-domain delayed discrete component,            giving rise to the multiplied frequency-domain delayed            discrete components; and    -   c) at least one summation unit for summing said multiplied        frequency-domain delayed discrete components with the        corresponding frequency-domain substantially non-delayed        discrete components, giving rise to an output frequency-domain        complex signal.

According to an embodiment of the present invention, the one or morecoefficients are at least one of the following:

-   -   a) phase coefficients; and    -   b) gain coefficients.

According to an embodiment of the present invention, thefrequency-domain transformation is a Fourier transform.

According to another embodiment of the present invention, the Fouriertransform is the FFT (Fast Fourier Transform).

According to still another embodiment of the present invention, aninverse frequency-domain transformation is applied on the outputfrequency-domain complex signal for obtaining an output time-domaincomplex signal.

According to still another embodiment of the present invention, theinverse frequency-domain transformation is the IFFT (Inverse FFT).

According to a further embodiment of the present invention, the systemfurther comprises a processing unit configured to calculate a time delaybetween two or more sampling channels.

According to still a further embodiment of the present invention, theoutput frequency-domain complex signal has a predefined frequencyspectrum that comprises one or more predefined frequencies, which arewhole multiples and/or half-multiples of a sampling frequency, accordingto which the analog signal is sampled.

According to another embodiment of the present invention, a system isconfigured to perform a complex sampling of a signal in afrequency-domain, said system comprising:

-   -   a) a non-delayed sampling channel module configured to provide a        plurality of frequency-domain substantially non-delayed discrete        signal components; and    -   b) one or more additional sampling channel modules, each        additional sampling channel module comprising at least one delay        unit and at least one coefficient unit for enabling providing a        plurality of frequency-domain delayed discrete components, said        plurality of frequency-domain delayed discrete components being        adapted to a specific frequency band, wherein said system is        further configured to combine the delayed discrete components        with the corresponding substantially non-delayed discrete        components, for generating an output frequency-domain complex        signal.

According to another embodiment of the present invention, thecoefficient unit provides coefficients for the specific frequency band.

According to still another embodiment of the present invention, a systemis configured to perform a complex sampling of a signal in a time-domainby means of a predefined-order sampling, said system comprising:

-   -   a) a sampling channel comprising:        -   a.1. at least one analog-to-digital converter configured to            convert an analog signal to a corresponding time-domain            substantially non-delayed digital signal;    -   b) one or more additional sampling channels enabling to perform        a predefined-order sampling, the predefined-order depending on a        number of said one or more additional sampling channels, each        additional sampling channel comprising:        -   b.1. at least one delay unit configured to delay an analog            signal by a predefined value, giving rise to a delayed            analog signal;        -   b.2. at least one analog-to-digital converter configured to            convert said delayed analog signal to a corresponding            delayed digital signal; and        -   b.3. at least one complex digital filter to be applied to            said delayed digital signal for generating complex samples            of said delayed digital signal, giving rise to a complex            time-domain delayed digital signal; and    -   c) at least one summation unit for combining the real portion of        said complex time-domain delayed digital signal with said        time-domain substantially non-delayed digital signal, giving        rise to a combined digital signal, and thereby giving a rise to        an output time-domain complex signal, the real portion of which        is said combined digital signal and the imaginary portion of        which is the imaginary portion of said complex time-domain        delayed digital signal.

According to still another embodiment of the present invention, thedigital filter is a FIR (Finite Impulse Response) filter.

According to a further embodiment of the present invention, a system isconfigured to perform a complex sampling of a signal in a time-domain,said system comprising:

-   -   a) a non-delayed sampling channel module configured to provide a        time-domain substantially non-delayed digital signal; and    -   b) one or more additional sampling channel modules, each        additional sampling channel module comprising at least one delay        unit and at least one complex digital filter unit for enabling        providing a complex time-domain delayed digital signal, wherein        the real portion of said complex time-domain delayed digital        signal is further combined with said time-domain substantially        non-delayed digital signal, giving rise to a combined digital        signal, and thereby giving a rise to an output time-domain        complex signal, the real portion of which is said combined        digital signal and the imaginary portion of which is the        imaginary portion of said complex time-domain delayed digital        signal.

According to an embodiment of the present invention, a method ofperforming complex sampling of a signal in a frequency-domain by meansof a predefined-order sampling, said method comprising:

-   -   a) providing a sampling channel configured to:        -   a.1. convert an analog signal to a corresponding            substantially non-delayed digital signal; and        -   a.2. transform said digital signal to a plurality of            corresponding frequency-domain substantially non-delayed            discrete components;    -   b) providing one or more additional sampling channels enabling        to perform a predefined-order sampling, the predefined-order        depending on a number of said one or more additional sampling        channels, each additional sampling channel configured to:        -   b.1. delay an analog signal by a predefined value, giving            rise to a delayed analog signal;        -   b.2. convert said delayed analog signal to a corresponding            delayed digital signal;        -   b.3. transform said delayed digital signal to a plurality of            frequency-domain delayed discrete components;        -   b.4. provide one or more corresponding coefficients for each            frequency-domain delayed discrete component; and        -   b.5. multiply said one or more corresponding coefficients            with said each corresponding frequency-domain delayed            discrete component, giving rise to the multiplied            frequency-domain delayed discrete components; and    -   c) combining said multiplied frequency-domain delayed discrete        components with the corresponding frequency-domain substantially        non-delayed discrete components, giving rise to an output        frequency-domain complex signal.

According to another embodiment of the present invention, a method ofperforming a complex sampling of a signal in a frequency-domain, saidmethod comprising:

-   -   a) generating a plurality of frequency-domain substantially        non-delayed discrete signal components;    -   b) generating a plurality of frequency-domain delayed discrete        components, said plurality of frequency-domain delayed discrete        components being adapted to a specific frequency band by means        of one or more corresponding coefficients; and    -   c) combining the delayed discrete components with the        corresponding substantially non-delayed discrete components, for        generating an output frequency-domain complex signal.

According to still another embodiment of the present invention, a methodof performing complex sampling of a signal in a time-domain by means ofa predefined-order sampling, said method comprising:

-   -   a) providing a sampling channel for converting an analog signal        to a corresponding time-domain substantially non-delayed digital        signal;    -   b) providing one or more additional sampling channels enabling        to perform a predefined-order sampling, the predefined-order        depending on a number of said one or more additional sampling        channels, each additional sampling channel configured to:        -   b.1. delay an analog signal by a predefined value, giving            rise to a delayed analog signal;        -   b.2. convert said delayed analog signal to a corresponding            delayed digital signal; and        -   b.3. generate complex samples of said delayed digital            signal, giving rise to a complex time-domain delayed digital            signal; and    -   c) combining the real portion of said complex time-domain        delayed digital signal with said time-domain substantially        non-delayed digital signal, giving rise to a combined digital        signal, and thereby giving a rise to an output time-domain        complex signal, the real portion of which is said combined        digital signal and the imaginary portion of which is the        imaginary portion of said complex time-domain delayed digital        signal.

According to still another embodiment of the present invention, themethod further comprises generating the complex time-domain delayeddigital signal by using a digital filter.

According to a further embodiment of the present invention, a method ofperforming a complex sampling of a signal in a time-domain, said methodcomprising:

-   -   a) generating a time-domain substantially non-delayed digital        signal;    -   b) generating a complex time-domain delayed digital signal; and    -   c) combining the real portion of said complex time-domain        delayed digital signal with said time-domain substantially        non-delayed digital signal, giving rise to a combined digital        signal, and thereby giving a rise to an output time-domain        complex signal, the real portion of which is said combined        digital signal and the imaginary portion of which is the        imaginary portion of said complex time-domain delayed digital        signal.

According to an embodiment of the present invention, a method ofcalculating a time delay between two or more sampling channels in asignal processing system, said method comprising:

-   -   a) providing a first sampling channel for enabling sampling of a        substantially non-delayed signal; and    -   b) providing one or more additional sampling channels, each        sampling channel providing a predefined delay τ to said signal,        giving rise to a delayed signal, and then enabling sampling of        said delayed signal, wherein said predefined delay τ is        calculated by using the relationship between said delay τ and        the phase difference Δφ of said delayed signal.

According to another embodiment of the present invention, the methodfurther comprises defining the relationship between the time delay τ andthe phase difference Δφ by means of at least one of the following:a) 2·π·f ₁·τ=Δφ₁+2·π·N; andb) 2·π·(f ₁ +Δf)·τ=Δφ₂·π·(N+M),wherein Δφ₁ is a phase difference of a first delayed signal having afirst predefined frequency f₁, Δφ₂ is a phase difference of a seconddelayed signal having a second predefined frequency f₂, Δf is adifference between said second and first predefined frequencies, therebyf₂=f₁+Δf, and M and N are predefined integers.

According to still another embodiment of the present invention, themethod further comprises determining the bound of integer M by using thefollowing relationship:

$M = {{{\tau \cdot \Delta}\; f} - {\frac{{\Delta\phi}_{2} - {\Delta\phi}_{1}}{2\pi}.}}$

According to still another embodiment of the present invention, themethod further comprises determining the approximation of the integer Mby considering that 0<Δφ₁<2π and 0<Δφ₂<2π.

According to still another embodiment of the present invention, themethod further comprises determining the approximation of the integer Mby considering that Δf is predefined.

According to a further embodiment of the present invention, the methodfurther comprises measuring frequency differences Δf₁₂ and Δf₁₃ betweenthe first predefined frequency f₁, the second predefined frequency f₂and a third predefined frequency f₃, giving rise to frequencydifferences Δf₁₂=f₂−f₁ and Δf₁₃=f₃−f₁.

According to still a further embodiment of the present invention, themethod further comprises calculating the time delay τ approximation byusing the one or more of the following:

$\begin{matrix}{{\tau = {\frac{{\Delta\phi}_{2} - {\Delta\phi}_{1}}{2{\pi \cdot \Delta}\; f_{12}} + \frac{M_{1}}{\Delta\; f_{12}}}};{and}} & \left. a \right) \\{{\tau = {\frac{{\Delta\phi}_{3} - {\Delta\phi}_{1}}{2{\pi \cdot \Delta}\; f_{13}} + \frac{M_{2}}{\Delta\; f_{13}}}},} & \left. b \right)\end{matrix}$

-   -   wherein Δφ₃ is a phase difference of a third delayed signal        having a third predefined frequency f₃, and M₁ and M₂ are        integers.

According to still a further embodiment of the present invention, themethod further comprises using the calculated time delay τ approximationfor determining a value of the integer M.

According to still a further embodiment of the present invention, themethod further comprises determining a value of the integer N by usingthe determined value of the integer M.

According to still a further embodiment of the present invention, themethod further comprises calculating the time delay τ by using bothdetermined values of the integers M and N.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to understand the invention and to see how it may be carriedout in practice, various embodiments will now be described, by way ofnon-limiting examples only, with reference to the accompanying drawings,in which:

FIG. 1A schematically illustrates a conventional interpolation system ofsecond order sampling, according to the prior art;

FIG. 1B schematically illustrates a conventional complex samplingsystem, in which an input signal is sampled in two sampling channels,while shifting the phase by ninety degrees, according to the prior art;

FIG. 2 is a schematic illustration of complex sampling in a frequencydomain by performing second-order sampling, according to an embodimentof the present invention;

FIG. 3 is a schematic illustration of a complex sampling system,performing sampling in a time domain, according to another embodiment ofthe present invention; and

FIG. 4 is a schematic illustration of a system for complex sampling byperforming sampling of 2M-order, according to still another embodimentof the present invention.

It will be appreciated that for simplicity and clarity of illustration,elements shown in the figures have not necessarily been drawn to scale.For example, the dimensions of some of the elements may be exaggeratedrelative to other elements for clarity. Further, where consideredappropriate, reference numerals may be repeated among the figures toindicate corresponding or analogous elements.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Unless specifically stated otherwise, as apparent from the followingteachings, it is noted that throughout the specification utilizing termssuch as “processing”, “computing”, “calculating”, “determining”, or thelike, refer to the action and/or processes of a computer (machine) thatmanipulate and/or transform data into other data, said data representedas physical, e.g. such as electronic, quantities. The term “computer”should be expansively construed to cover any kind of electronic devicewith data processing capabilities, comprising, by the way ofnon-limiting examples, personal computers, servers, computingsystems/units, communication devices, processors (e.g., digital signalprocessors (DSPs), microcontrollers, field programmable gate arrays(FPGAs), application specific integrated circuits (ASICs), etc.), andany other electronic computing devices. Also it should be noted thatoperations in accordance with the teachings herein may be performed by acomputer that is specially constructed for the desired purposes or by ageneral purpose computer that is specially configured for the desiredpurpose by means of a computer program stored in a computer readablestorage medium.

FIG. 2 is a schematic illustration 200 of complex sampling in afrequency domain by performing second-order sampling, according to anembodiment of the present invention. According to this embodiment, inputsignal X(t) is first filtered by means of filter 151 in order to removethe undesired frequency range (in a time domain). Then, the filteredsignal X₁(t) is sampled by means of two sampling channels(systems/modules) 205′ and 205″, having a predefined time delay τbetween them, and then is converted to corresponding digital signals bymeans of conventional A/D converters 105′ and 105″, giving rise to X₁(l)and X₂(l) signals respectively. After that, digital signals X₁(l) andX₂(l) are processed and converted to a frequency domain by means of theFFT (Fast Fourier Transform), which is a conventional technique forperforming a discrete Fourier transform. As a result, discrete signalsX₁′(k) and X₂′(k) are obtained, wherein k is an index. It should benoted that the frequency band of input analog signal X(t) is known (thefrequency bandwidth is equal to the sampling frequency Fs), andtherefore the phase difference of each frequency component of thedelayed signal X₂′(k), which is provided via delayed sampling channel205″, can be calculated. According to an embodiment of the presentinvention, the frequency spectrum of delayed signal X₂′(k) is multipliedby a corresponding set of predefined phase and gain coefficients Q(k)250, each having a gain and phase (e.g., Q(k)=g_(k)·exp(i·Δφ_(k)),wherein k is an index). Then, the spectrum of delayed signal X₂′(k) issummed (combined) with the spectrum of signal X₁′(k), giving rise to(generating) signal Y(k) that has a spectrum being equivalent to thespectrum of a complex signal, which can be also obtained by means ofconventional complex sampling.

According to an embodiment of the present invention, the desired timedelay τ may be different for different frequency bands. Further, thetime delay τ may be obtained either by providing time delaycomponent/unit 103 (in which the time delay τ can be predefined) or byperforming a phase difference (e.g., a phase shift) of a samplingfrequency, leading to a desired time delay of a signal. It should benoted that one or more phase and gain coefficients Q(k) 250 are used for(are applied to) each frequency component of signal X₂′(k). These phaseand gain coefficients Q(k), provided within the correspondingcoefficients data unit 250, can be predefined, for example, empiricallyby substantially accurate measuring of the above time delay r. It shouldbe noted that even in a case when time delay τ is a frequency-dependentcomponent, the corresponding phase and gain coefficients Q(k) can bestill calculated and predefined thereof.

According to an embodiment of the present invention, the phase and gaincoefficients are pre-calculated during the calibration process of system200, and then are stored within the memory means (not shown), whilethere is a need for a coefficient for each frequency component of signalX₂′(k), after applying the FFT transform. Further, for calculating thecorresponding phase difference Δφ(k), there is a need to provide asignal of a predefined frequency, and then calculate the correspondingphase difference Δφ(k) between the delayed and reference signals X₂′(k)and X′(k), respectively. In addition, it should be noted that the powerratio between two channels (the non-delayed channel 205′ and delayedchannel 205″) is calculated and corresponding gain coefficients g, (k isan index) are determined and stored within memory means (not shown) forlater usage. This can be achieved in several ways, according to variousembodiments of the present invention. According to one embodiment of thepresent invention, substantially all frequencies that correspond to theFFT frequency component to be calculated are provided, and then a phasedifference for each such component is calculated. If it is supposed, forexample, that the frequency range is (F_(s), 2F_(s)) and the FFT lengthis N, then the set of frequencies f(k) that correspond to the FFTfrequency components are:

$\begin{matrix}{{{f(k)} = {F_{s} + {\frac{k}{N} \cdot F_{s}}}},{{{for}\mspace{14mu} k} = 0},\ldots\;,{N - 1}} & (1)\end{matrix}$

For each of the above N frequencies, the phase difference Δφ between thesampling channels 205′ and 205″ is calculated.

According to another embodiment of the present invention, a number offrequencies are provided in intervals that are greater than the FFT bin(bin is defined as F/N, wherein N represents a number of FFT frequencycomponents), and then the phase difference for each provided frequencyis calculated by performing interpolation for each FFT frequencycomponent. Thus, for example, if the frequency range is (F_(s), 2F_(s))and N/16 frequencies are provided, then the set of frequencies thatcorrespond to the FFT frequency components are:

$\begin{matrix}{{{f(m)} = {{Fs} + {\frac{16m}{N} \cdot {Fs}}}},{{{for}\mspace{14mu} m} = 0},\ldots\;,{{N\text{/}16} - 1}} & (2)\end{matrix}$

According to this embodiment, for each of the above N/16 frequencies,the phase difference Δφ between the sampling channels 205′ and 205″ iscalculated. After that, the phase differences for each frequencycomponent is calculated by performing interpolation of the correspondingphase difference Δφ for each FFT frequency component of signal X₂′(k):

$\begin{matrix}{{\Delta\;{\varphi\left( {k;{{16m} \leq k < {{16m} + 15}}} \right)}} = {{\Delta\;{\phi(m)}} + {\frac{k - {16m}}{16}\left( {{{\Delta\phi}\left( {m + 1} \right)} - {{\Delta\phi}(m)}} \right)}}} & (3)\end{matrix}$

According to a further embodiment of the present invention, a relativelysmall number of frequencies are provided in non-uniform frequencyintervals, and then the phase differences Δφ(m) between thesefrequencies are determined. After that, the time delay τ is calculatedby using the above-determined phase differences Δφ(m) by means of anovel method for calculating time delays between sampling channels (suchas channels 205′ and 205″), according to an embodiment of the presentinvention. The phase differences Δφ(m) for each FFT frequency componentcan be calculated by using the following equation:Δφ=mod(2π·f·τ)_(2π)  (4)wherein f is a frequency, and mod(·)_(2π) is a “modulo” mathematicaloperator.

In general, when a real signal (having frequency f) is received, and theFFT (having length N) of said signal is calculated, then the resultingsignal will appear at the frequency spectrum as a frequency component ofthe FFT bin k (generally, a FFT bin is a single frequency of the FFT, towhich each frequency component contributes):

$\begin{matrix}{f = {F_{s} \cdot \left( {n + \frac{k}{N}} \right)}} & (5)\end{matrix}$wherein f is a signal frequency; Fs is a sampling frequency; N is theFFT length; n is an integer; and k is a FFT bin number. Also, anadditional (for example, undesired) frequency component appears in theFFT bin (N−k), due to the symmetry of the conventional FFT. It should benoted that the phase difference Δφ_(N-k) of the above undesiredfrequency component has an opposite sign compared to the phasedifference Δφ_(k) of the desired frequency component that appears in theFFT bin k.

According to an embodiment of the present invention, in order to cancelthe above undesired frequency component appearing in the FFT bin (N−k),the frequency components can be summed (combined) by using the followingequation:Y(k)=X ₁′(k)−g _(N-k)·exp(i·Δφ _(N-k))·X ₂′(k)  (6)wherein Δφ_(N-k) is a predefined phase difference of the frequencycomponent that contributes to bin N−k; g_(N-k) is a gain coefficientcalculated for that frequency component; X₁′(k) and X₂′(k) arecorresponding frequency components of non-delayed and delayed signals,respectively; and Y(k) represents frequency components at the output ofsystem 200. As a result, the frequency spectrum of the output signalY(k) is equivalent to the frequency spectrum of the conventional complexsampling.

It should be noted that according to an embodiment of the presentinvention, when a complex sampling is required in a time domain, then aninverse frequency-domain transformation, such as the Inverse Fouriertransform (IFFT), can be performed on the frequency spectrum obtained byimplementing system 200.

According to an embodiment of the present invention, time delays (timedifferences) between sampling channels (such as channels 205′ and 205″(FIG. 2)) can be calculated in a relatively accurate manner (forexample, by means of a processing unit/system), as described below indetail. According to this embodiment, the corresponding time delay τ canbe calculated by using the following relationship between the time delayτ and phase difference Δφ:2·π·f ₁·τ=Δφ₁+2·π·N  (7)and2·π·(f ₁ +Δf)·τ=Δφ₂+2·π·(N+M)  (8)wherein Δφ₁ is a phase difference of a signal having frequency f₁; Δφ₂is a phase difference of a signal having frequency f₂, while f₂=f₁+Δf;and M and N are integers. It should be noted that the above twoequations have three variables: time delay τ, integer M and integer N.

For the bound range of values of time delay τ, the bound for integer Mcan be determined by using the following equation, which is a result ofsubtracting equation (8) from equation (7):

$\begin{matrix}{M = {{{\tau \cdot \Delta}\; f} - \frac{{\Delta\;\phi_{2}} - {\Delta\;\phi_{1}}}{2\pi}}} & (9)\end{matrix}$

In such a way, the first approximation of integer M can be determined,considering that 0<Δφ₁<2π and 0<Δφ₂<2π, and considering that Δf isknown.

Further, by measuring frequency differences Δf₁₂ and Δf₁₃ between threepredefined frequencies f₁, f₂ and f₃, such that Δf₁₂=f₂−f₁ andΔf₁₃=f₃−f₁, the corresponding time delay τ can be calculated by dividingthe above equation (9) by said frequency differences Δf₁₂ and Δf₁₃,respectively:

$\begin{matrix}{\tau = {\frac{{\Delta\phi}_{2} - {\Delta\phi}_{1}}{2{\pi \cdot \Delta}\; f_{12}} + {\frac{M_{1}}{\Delta\; f_{12}}\mspace{14mu}{and}}}} & (10) \\{\tau = {\frac{{\Delta\phi}_{3} - {\Delta\phi}_{1}}{2{\pi \cdot \Delta}\; f_{13}} + \frac{M_{2}}{\Delta\; f_{13}}}} & (11)\end{matrix}$wherein M₁ and M₂ are bounded integers. Then, as a result, the followingequation is obtained:

$\begin{matrix}{{\frac{{\Delta\phi}_{2} - {\Delta\phi}_{1}}{2{\pi \cdot \Delta}\; f_{12}} + \frac{M_{1}}{\Delta\; f_{12}}} = {\frac{{\Delta\phi}_{3} - {\Delta\phi}_{1}}{2{\pi \cdot \Delta}\; f_{13}} + \frac{M_{2}}{\Delta\; f_{13}}}} & (12)\end{matrix}$

Thus, considering that M₁ and M₂ are bounded integers, and also phasedifferences Δφ₁, Δφ₂, Δφ₃ and frequency differences Δf₁₂, Δf₁₃ are allknown, then the first approximation of time delay τ can be determined.This time delay approximation can be inserted in equation (9) forobtaining a value of M in a relatively accurate manner, considering thatM is a bounded integer. Then, after determining the value of M, thevalue of N can be also determined by inserting the determined value of Minto equations (7) and (8). As a result, both bounded integers M and Nare determined, and the time delay τ is calculated in a relativelyaccurate manner by using the same equations (7) and (8).

It should be noted that according to an embodiment of the presentinvention, the range of time delays r can be selected in the followingway. It is supposed, for example, that the frequencies are within therange of [F_(start), F_(start)+BW], wherein F, is a starting frequency,and BW is a bandwidth, while Fs≧BW (Fs is a sampling frequency). Thegain (in dB (Decibels)) for the desired frequency component (of the FFT)can be presented by the following equation:

$\begin{matrix}{10\mspace{11mu}{\log_{10}\left( {2{{\sin\left( \frac{{\Delta\;\varphi_{k}} + {\Delta\varphi}_{N - k}}{2} \right)}}^{2}} \right)}} & (13)\end{matrix}$wherein Δφ_(k) is the phase difference of the frequency component thatappears in the FFT bin k, when the frequency is

${f = {\left( {n + \frac{k}{N}} \right) \cdot F_{s}}};$and Δφ_(N-k) is the phase difference of the frequency component thatappears in the FFT bin N−k, when the frequency is

$f = {\left( {\overset{\sim}{n} + \frac{N - k}{N}} \right) \cdot {F_{s}.}}$If nF_(s)≧F_(start)>(n−1)·F_(s), then the phase difference Δφ_(k) ispresented by:

$\begin{matrix}{{\Delta\varphi}_{k} = {2{{\pi\left( {n + \frac{k}{N}} \right)} \cdot F_{s} \cdot \tau}}} & (14)\end{matrix}$wherein n is an integer, and τ is a time delay, which can be, forexample, in the range determined by the following equation:

$\begin{matrix}{\frac{5}{6 \cdot \left( {{2n} + 1} \right)} > {F_{s} \cdot \tau} > \frac{1}{6 \cdot \left( {{2n} - 1} \right)}} & (15)\end{matrix}$

It should be noted that selecting the delay τ within the above rangeensures that in addition to removing the undesired frequency component(FFT bin (N−k)) of the 15 frequency spectrum, the power of the desiredfrequency component (FFT bin k) will not be decreased more than 3 dB(Decibels), as shown in the equation below:

$\begin{matrix}{{10\mspace{14mu}{\log_{10}\left( {2{{\sin\left( \frac{{\Delta\;\varphi_{k}} + {\Delta\varphi}_{N - k}}{2} \right)}}^{2}} \right)}} > {{- 3}\mspace{11mu}{dB}}} & (16)\end{matrix}$wherein Δφ_(k) and Δφ_(N-k) are phase differences in bins k and (N−k),respectively. In addition, it should be noted that any other constraintscan be considered, such as ensuring that the power of the desiredfrequency component will not be decreased, for example, more than 2 dB(instead of 3 dB), and the like.

FIG. 3 is a schematic illustration of a complex sampling system 300,performing sampling in a time domain, according to another embodiment ofthe present invention. According to this embodiment of the presentinvention, signal X₂(l) passes through a digital FIR (Finite ImpulseResponse) filter unit 310. This filter is a complex filter and at itsoutput, complex signal samples are obtained. The real part of the signalsamples after FIR filter 310 is added to signal X₁(l) that is outputtedfrom A/D converter 105′, giving rise to Re{Y(s)} signal, which is a realpart of the signal, to which the complex sampling is applied. On theother hand, in the delayed sampling channel 305″, the imaginary part ofthe signal samples, after passing via the FIR filter 310, is theimaginary part (Im{Y(s)}) of the signal, to which the complex samplingis applied.

According to an embodiment of the present invention, the FIR filtercoefficients h(p) can be obtained by applying an inverse Fast Fouriertransform (IFFT) on phase and gain coefficients Q(k) 250 (FIG. 2):

$\begin{matrix}{{h(p)} = {\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{g_{k} \cdot {\exp\left( {{{\mathbb{i}} \cdot \Delta}\;\varphi_{k}} \right)} \cdot {\exp\left( {2{\pi \cdot {\mathbb{i}}}\frac{k \cdot p}{N}} \right)}}}}} & (17)\end{matrix}$wherein g_(k) and Δφ_(k) are a gain and phase difference, respectively,of the corresponding signal passing via delayed sampling channel 305″; kand p are indices; i is √{square root over (−1)}; and N is a number offrequency components. It should be noted that each phase and gain 20coefficient Q(k) can be equal to g_(k)·exp(i·Δφ_(k)), which is indicatedwithin the above expression of h(p).

FIG. 4 is a schematic illustration of a system 400 for complex samplingby performing sampling of the 2M-order sampling (the predefined-ordersampling), according to still another embodiment of the presentinvention. According to this embodiment, if providing 2M samplingchannels (two or more sampling channels) with predefined delays τ₁, τ₂,. . . , τ_(n) between them, then operating with signal bandwidthBW=M·F_(s) becomes possible, wherein F_(s) is a sampling frequency. Thiscan be compared to system 200 (FIG. 2), for which signal bandwidth BW isless or equal to the sampling frequency: BW≦F_(s).

It is supposed, for example, that sampling channels 205′, 205″, etc. arerepresented by index n, while nε[1, 2M]. The output frequency bands(Band 1, Band 2, etc.) are represented by index m, while mε[1, M]. Inaddition, each FFT bin is numbered by index k. The FFT of a signal iscalculated in each sampling channel and is represented as X_(n)(k), theoutput frequency spectrum is represented as Y_(m)(k), and phase and gaincoefficients 250′ are shown as Q_(m) ^(n)(k). Thus, according to anembodiment of the present invention, the output frequency signalY_(m)(k) can be calculated by using the following equation, in whicheach at least one phase and gain coefficient Q_(m) ^(n)(k) is multipliedwith its corresponding signal X_(n)(k):

$\begin{matrix}{{Y_{m}(k)} = {\sum\limits_{n = 1}^{2M}{{Q_{m}^{n}(k)} \cdot {X_{n}(k)}}}} & (18)\end{matrix}$

It can be further supposed, for example, that input frequency F belongsto Band M if Fε[F_(start)+(m−1)F_(s), F_(start)+m·F_(s)], whereinF_(start) is a starting frequency that is defined manually orautomatically according to the need of a user of system 400; and F_(s)is a sampling frequency, while mε[1, M]. Also, the frequency appears inthe FFT bin, if one of the following two equations takes place:

$\begin{matrix}{{{{round}\left( {{{mod}\left( {F,F_{s}} \right)} \cdot \frac{N}{F_{s}}} \right)} = {k\mspace{20mu}{or}}}\mspace{14mu}{{{round}\left( {{{mod}\left( {F,F_{s}} \right)} \cdot \frac{N}{F_{s}}} \right)} = \left( {N - k} \right)}} & (19)\end{matrix}$wherein k and (N−k) are corresponding FFT bins; N is the FFT length; andmod(·) is a “modulo” mathematical operator.

The phase difference Δφ_(n) ^(m)(k) of each corresponding frequencycomponent depends on frequency F^(m)(k) (of FFT bin k in Band m (mε[1,M])) and on the sampling channel delay τ₁, τ₂, . . . , τ_(n), as shownin the following equation:Δφ_(n) ^(m)(k)=2π·F ^(m)(k)·τ_(n)  (20)

It should be noted that the frequency spectrum of a signal X_(n)(k)passing via each corresponding sampling channel (such as samplingchannels 205′, 205″, etc.) is composed of frequencies received from allbands (such as Band 1, Band 2, etc.). Thus, signals from 2M possiblefrequency sources are provided to the corresponding bin k of the FFT, aspresented in the following equation.

$\begin{matrix}{{X_{n}(k)} = {\sum\limits_{m = 1}^{M}\left\lbrack {{{X\left( {F^{m}(k)} \right)} \cdot {\mathbb{e}}^{{\mathbb{i}} \cdot {{\Delta\varphi}_{n}^{m}{(k)}}}} + {{X^{*}\left( {F^{m}\left( {N - k} \right)} \right)} \cdot {\mathbb{e}}^{{- {\mathbb{i}}} \cdot {{\Delta\varphi}_{n}^{m}{({N - k})}}}}} \right\rbrack}} & (21)\end{matrix}$

As a result, the corresponding matrices of signals X_(n)(k) can bepresented as follows:

$\begin{matrix}{\begin{pmatrix}{X_{1}(k)} \\{X_{2}(k)} \\\vdots \\{X_{{2M} - 1}(k)} \\{X_{2M}(k)}\end{pmatrix} = {\begin{pmatrix}{\mathbb{e}}^{{\mathbb{i}} \cdot {{\Delta\varphi}_{1}^{1}{(k)}}} & {\mathbb{e}}^{{- {\mathbb{i}}} \cdot {{\Delta\varphi}_{1}^{1}{({N - k})}}} & \ldots & {\mathbb{e}}^{{- {\mathbb{i}}} \cdot {{\Delta\varphi}_{1}^{M}{({N - k})}}} \\{\mathbb{e}}^{{\mathbb{i}} \cdot {{\Delta\varphi}_{2}^{1}{(k)}}} & \vdots & \ddots & \vdots \\\vdots & \vdots & \ddots & \vdots \\{\mathbb{e}}^{{\mathbb{i}} \cdot {{\Delta\varphi}_{{2M} - 1}^{1}{(k)}}} & \vdots & \ddots & \vdots \\{\mathbb{e}}^{{\mathbb{i}} \cdot {{\Delta\varphi}_{2M}^{1}{(k)}}} & {\mathbb{e}}^{{- {\mathbb{i}}} \cdot {{\Delta\varphi}_{2M}^{1}{({N - k})}}} & \ldots & {\mathbb{e}}^{{- {\mathbb{i}}} \cdot {{\Delta\varphi}_{2M}^{M}{({N - k})}}}\end{pmatrix} \cdot \begin{pmatrix}{X\left( {F^{1}(k)} \right)} \\{X^{*}\left( {F^{1}\left( {N - k} \right)} \right)} \\\vdots \\{X\left( {F^{M}(k)} \right)} \\{X^{*}\left( {F^{M}\left( {N - k} \right)} \right)}\end{pmatrix}}} & (22)\end{matrix}$wherein k and (N−k) are corresponding FFT bins; and N is the FFT length.If it is supposed, for example, that P(k) matrix is defined as follows:

$\begin{matrix}{{P(K)} = \begin{pmatrix}{\mathbb{e}}^{{\mathbb{i}} \cdot {{\Delta\varphi}_{1}^{1}{(k)}}} & {\mathbb{e}}^{{- {\mathbb{i}}} \cdot {{\Delta\varphi}_{1}^{1}{({N - k})}}} & \ldots & {\mathbb{e}}^{{- {\mathbb{i}}} \cdot {{\Delta\varphi}_{1}^{M}{({N - k})}}} \\{\mathbb{e}}^{{\mathbb{i}} \cdot {{\Delta\varphi}_{2}^{1}{(k)}}} & \vdots & \ddots & \vdots \\\vdots & \vdots & \ddots & \vdots \\{\mathbb{e}}^{{\mathbb{i}} \cdot {{\Delta\varphi}_{{2M} - 1}^{1}{(k)}}} & \vdots & \ddots & \vdots \\{\mathbb{e}}^{{\mathbb{i}} \cdot {{\Delta\varphi}_{2M}^{1}{(k)}}} & {\mathbb{e}}^{{- {\mathbb{i}}} \cdot {{\Delta\varphi}_{2M}^{1}{({N - k})}}} & \ldots & {\mathbb{e}}^{{- {\mathbb{i}}} \cdot {{\Delta\varphi}_{2M}^{M}{({N - k})}}}\end{pmatrix}} & (23)\end{matrix}$then, by further considering that the desired frequency spectrum at theoutput is

$\left. {{Y_{m}(k)} = {X\left( {F^{m}(k)} \right)}} \right) = {\sum\limits_{n = 1}^{2M}{{Q_{m}^{n}(k)} \cdot {X_{n}(k)}}}$(i.e., the output signal Y_(m)(k) is adapted to a specific frequencyband/spectrum), the corresponding phase and gain coefficients Q_(m)^(n)(k) can be calculated by inverting the matrix P(k) and obtaining:

$\begin{matrix}{{Q_{m}^{n}(k)} = {\begin{pmatrix}1 & 0 & \ldots & 0 & 0 \\\vdots & \ddots & \ddots & \ddots & \vdots \\0 & 0 & \ldots & 1 & 0\end{pmatrix} \cdot {P^{- 1}(k)}}} & (24)\end{matrix}$

Thus, for example, if M=2, the phase and gain coefficients Q_(m) ^(n)(k)are equal to:

$\begin{matrix}{{Q_{m}^{n}(k)} = {\begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & 1 & 0\end{pmatrix} \cdot {P^{- 1}(k)}}} & (25)\end{matrix}$

For another example, if M=3, then the phase and gain coefficients Q_(m)^(n)(k) are equal to:

$\begin{matrix}{{Q_{m}^{n}(k)} = {\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0\end{pmatrix} \cdot {P^{- 1}(k)}}} & (26)\end{matrix}$

According to an embodiment of the present invention, the constraint forselecting time delay values in this case can be such that P(k) matrix isnot singular, which means that the determinant of said P(k) matrix doesnot become equal to zero or almost equal to zero (i.e., there are no twoor more substantially equal time delays τ, for example).

While some embodiments of the invention have been described by way ofillustration, it will be apparent that the invention can be put intopractice with many modifications, variations and adaptations, and withthe use of numerous equivalents or alternative solutions that are withinthe scope of persons skilled in the art, without departing from thespirit of the invention or exceeding the scope of the claims.

The invention claimed is:
 1. A method of calculating a time delaybetween two or more sampling channels in a signal processing system,said method comprising: a) providing a first sampling channel forenabling sampling of a substantially non-delayed signal; and b)providing one or more additional sampling channels, each samplingchannel providing a predefined delay τ to said signal, giving rise to adelayed signal, and then enabling sampling of said delayed signal,wherein said predefined delay τ is calculated by using the relationshipbetween said delay τ and a phase difference Δφ of said delayed signal;c) defining the relationship between the time delay τ and the phasedifference Δφ by means of at least one of the following:a) 2·π·f ₁·τ=Δφ₁+2·π·N; andb) 2·π·(f ₁ +Δf)·τ=Δφ₂+2·π·(N+M), wherein Δφ₁ is a phase difference of afirst delayed signal having a first predefined frequency f₁, Δφ₂ is aphase difference of a second delayed signal having a second predefinedfrequency f₂ , Δf is a difference between said second and firstpredefined frequencies, thereby f₂=f₁+Δf, and M and N are integers; d)determining a bound of integer M by using the following relationship:${M = {{{\tau \cdot \Delta}\; f} - \frac{{\Delta\;\phi_{2}} - {\Delta\;\phi_{1}}}{2\pi}}};$e) measuring frequency differences Δf₁₂ and Δf₁₃ between the firstpredefined frequency f₁, the second predefined frequency f₂ and a thirdpredefined frequency f₃, giving rise to frequency differences Δf₁₂=f₂−f₁and Δf₁₃=f₃−f₁;and f) calculating the time delay τ approximation byusing the one or more of the following: $\begin{matrix}{{\tau = {\frac{{\Delta\;\phi_{2}} - {\Delta\;\phi_{1}}}{2{\pi \cdot \Delta}\; f_{12}} + \frac{M_{1}}{\Delta\; f_{12}}}};{and}} & \left. a \right) \\{{\tau = {\frac{{\Delta\;\phi_{3}} - {\Delta\;\phi_{1}}}{2{\pi \cdot \Delta}\; f_{13}} + \frac{M_{2}}{\Delta\; f_{13}}}},} & \left. b \right)\end{matrix}$ wherein Δφ₃, is a phase difference of a third delayedsignal having the third predefined frequency f₃ , and M₁ and M₂ areintegers.
 2. The method according to claim 1, further comprisingdetermining an approximation of the integer M by considering that0<Δφ₁<2π and 0<Δφ₂<2π.
 3. The method according to claim 2, furthercomprising determining the approximation of the integer M by using Δf.4. The method according to claim 1, further comprising using thecalculated time delay τ approximation for determining a value of theinteger M.
 5. The method according to claim 4, further comprisingdetermining a value of the integer N by using the determined value ofthe integer M.
 6. The method according to claim 5, further comprisingcalculating the time delay τ by using both determined values of theintegers M and N.